REE Board Exam cheat sheet

Quadratic formula: for ax^2 + bx + c = 0, the roots are x = [-b ± sqrt(b^2 - 4ac)]/(2a); the discriminant b^2 - 4ac determines whether roots are real distinct, real equal, or complex.

Law of sines: a/sin A = b/sin B = c/sin C = 2R; law of cosines: c^2 = a^2 + b^2 - 2ab cos C.

Derivative rules: d/dx(x^n) = n x^(n-1); d/dx(e^x) = e^x; d/dx(ln x) = 1/x; apply product, quotient, and chain rules for composites.

Integral rules: ∫x^n dx = x^(n+1)/(n+1) + C for n ≠ -1; ∫(1/x) dx = ln|x| + C; the definite integral equals F(b) - F(a) for an antiderivative F.

Probability rules: P(A ∪ B) = P(A) + P(B) - P(A ∩ B); independent events satisfy P(A ∩ B) = P(A)P(B); conditional probability P(A|B) = P(A ∩ B)/P(B).

Complex numbers: z = a + bi has modulus |z| = sqrt(a^2 + b^2) and argument θ = atan2(b, a); Euler form is z = |z|e^(iθ).

2x2 matrix inverse: for A = [[a, b], [c, d]], A^-1 = (1/(ad - bc))[[d, -b], [-c, a]], provided the determinant ad - bc is nonzero.

Geometry formulas: circle area = πr^2 and circumference = 2πr; sphere volume = (4/3)πr^3 and surface area = 4πr^2; triangle area = (1/2)ab sin C.

Newton's second law: F_net = ma; weight is W = mg, using standard gravity g = 9.80665 m/s^2 near Earth's surface.

Work-energy relations: work W = Fd cos θ; kinetic energy KE = (1/2)mv^2; gravitational potential energy PE = mgh; total mechanical energy is conserved without nonconservative work.

Ideal gas law: PV = nRT, with R = 8.314 J/(mol·K); combined gas law P1V1/T1 = P2V2/T2 requires absolute temperature in kelvin.

First law of thermodynamics: ΔU = Q - W, where Q is heat added to the system and W is work done by the system.

Carnot efficiency: η_Carnot = 1 - T_C/T_H, where T_C and T_H are absolute cold and hot reservoir temperatures in kelvin.

Statics conditions: a rigid body in equilibrium satisfies ΣF = 0 and ΣM = 0 about any point; friction satisfies f ≤ μN for static contact.

Stress-strain relations: normal stress σ = F/A; engineering strain ε = ΔL/L; within the elastic limit, Hooke's law gives σ = Eε.

Fluid mechanics: pressure at depth is P = P0 + ρgh; continuity for incompressible flow is A1v1 = A2v2; Bernoulli equation is P + (1/2)ρv^2 + ρgh = constant along a streamline.

AC sinusoidal relations: V_rms = V_peak/√2; average real power P = VI cos θ; reactive power Q = VI sin θ; apparent power S = VI; power factor = cos θ = P/S.

Impedance and reactance: Z = R + jX; X_L = 2πfL; X_C = 1/(2πfC); series resonance occurs at f = 1/(2π√(LC)).

Balanced three-phase: P = √3 V_LL I_L cos θ; Q = √3 V_LL I_L sin θ; S = √3 V_LL I_L; in wye connection V_LL = √3 V_LN.

Ideal transformer: V1/V2 = N1/N2; current ratio I1/I2 = N2/N1; apparent power is unchanged, so S1 = S2 in the ideal case.

Per-unit method: Z_pu = Z_actual/Z_base; Z_base = V_base^2/S_base; when changing bases, Z_pu,new = Z_pu,old (S_base,new/S_base,old)(V_base,old/V_base,new)^2.

Induction machine: synchronous speed N_s = 120f/P; slip s = (N_s - N_r)/N_s; rotor frequency f_r = s f.

Transmission line losses: three-phase copper loss is approximately 3I^2R; voltage drop per phase is approximately I(R cos φ + X sin φ) for a short line.

Power factor correction: capacitor reactive power Q_c = P(tan φ1 - tan φ2) raises power factor from cos φ1 to cos φ2; a balanced three-phase fault current is I_f = V_LN/Z1 = V_LL/(√3 Z1).